Arrange the following bases in increasing order of their basic strength
`NH_(4)OH , NaOH , Mg(OH)_(2), AI(OH)_(3) `

Explanation

To arrange the given bases in increasing order of their basic strength, we will analyze each compound based on their properties and the metallic character of the elements involved.

### Step-by-Step Solution:

1. **Identify the Bases**: The bases given are:
- NH₄OH (Ammonium hydroxide)
- NaOH (Sodium hydroxide)
- Mg(OH)₂ (Magnesium hydroxide)
- Al(OH)₃ (Aluminum hydroxide)

2. **Understand Basic Strength**: Basic strength of a hydroxide is generally influenced by the metallic character of the metal in the hydroxide. The more metallic the element, the stronger the base tends to be.

3. **Metallic Character**:
- Sodium (Na) is a highly metallic element.
- Magnesium (Mg) is less metallic than sodium but more than aluminum.
- Aluminum (Al) is less metallic than magnesium.
- The ammonium ion (NH₄⁺) is derived from a non-metal (nitrogen), making NH₄OH a weak base.

4. **Order of Basic Strength**:
- **NaOH**: Strong base due to high metallic character of sodium. It fully dissociates in solution.
- **Mg(OH)₂**: Moderate base; it is less basic than NaOH but more basic than Al(OH)₃.
- **Al(OH)₃**: Weak base; it does not fully dissociate in solution and has lower basic strength than Mg(OH)₂.
- **NH₄OH**: Weakest base among the four; it does not fully dissociate in solution and is derived from a non-metal.

5. **Arranging in Increasing Order**:
- The increasing order of basic strength is:
- NH₄OH < Al(OH)₃ < Mg(OH)₂ < NaOH

### Final Answer:
**Increasing order of basic strength**:
\[ \text{NH}_4\text{OH} < \text{Al(OH)}_3 < \text{Mg(OH)}_2 < \text{NaOH} \]